PAC-Bayesian Learning and Domain Adaptation
Pascal Germain1 Francois Laviolette1 Amaury Habrard2 Emilie Morvant3
1
GRAAL Machine Learning Research GroupDepartement d’informatique et de genie logiciel
Universite Laval, Quebec, Canada{pascal.germain,francois.laviolette}@ift.ulaval.ca
2Laboratoire Hubert Curien
Saint-Etienne University, [email protected]
3
Laboratoire d’Informatique FondamentaleQARMA Group
Aix-Marseille University, [email protected]
NIPS 2012 Workshop: Multi-Trade-offs in Machine Learning,December 7, 2012
PAC-Bayesian Learning and Domain Adaptation
Outline
1 Domain AdaptationProblem DescriptionA Classical Domain Adaptation BoundA New Domain Adaptation Bound
2 PAC-Bayesian LearningPAC-Bayesian Learning of Linear ClassifierPAC-Bayesian Domain Adaptation Learning of Linear Classifiers
3 Preliminary Experimental Results
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 2 / 10
Domain Adaptation (DA) : Problem Description
When we need DA
The Learning distribution is different from the Testing distribution.
An example of a DA problem
• We have labeled images from a Web image corpus
• Is there a Person in unlabeled images from a Video corpus ?
Person no Person
?
Is there a Person ?
⇒ How to learn, from the source domain, a low-error classifier on the target one ?
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 3 / 10
Domain Adaptation (DA) : Problem Description
Supervised Classification
• We consider binary classification task: X input space, Y = {−1, 1} label set
• PS source domain: distribution over X×Y ; DS marginal distribution over X
• S ∼ (PS)m a labeled source sample
=⇒ Objective: Find a classifier h ∈ H with a low source risk RPS (h) .
Domain Adaptation
• PT target domain: distribution over X×Y ; DT marginal distribution over X
• T ∼ (DT )m′
a unlabeled target sample
=⇒=⇒=⇒ Objective: Find a classifier h ∈ H with a low target risk RPT (h) .
Supervised Classification
Labeled Sample From P
ModelLearning
Distribution P
S
S
Unlabe led Sample From D
Do m a in Ad a p t a t io n
Labe led Sample From P
ModelLearning
Dis tribution P
S
S T
Diffe rent Dis tribution PT
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 4 / 10
A Classical Domain Adaptation Bound (VC-dim approach)
• Let H be an hypothesis space.
Theorem [Ben-David et al., 2010]
For every h ∈ H and for all δ ∈ ]0, 1], with probability at least 1− δ :
RPT(h) ≤ RPS
(h) + 12dH∆H(DS ,DT ) + λ,
with λ= minh∗∈H
(RPS
(h∗) + RPT(h∗)).
Trade-off between:� RPS (h) is the classical expected error on the source domain
� dH∆H(DS ,DT ) is the H∆H-distance between source and target domains
dH∆H(DS ,DT ) = 2 suph,h′∈H∆H
∣∣ Prx∼DS
(h(x) 6=h′(x))− Prx∼DT
(h(x) 6=h′(x))∣∣
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 5 / 10
A New Domain Adaptation Bound (PAC-Bayesian approach)
• Let H be an hypothesis space.
• Given a weight distribution ρ ∼ H, we study the ρ-average errors:
RPS(Gρ) = E
h∼ρRPS
(h) , RPT(Gρ) = E
h∼ρRPT
(h) .
Theorem
For all δ ∈ ]0, 1], with probability at least 1− δ, for every posterior distribution ρ:
Eh∼ρ
RPT(h) ≤ E
h∼ρRPS
(h) + disρ(DS ,DT ) + λρ,
with λρ=RPS(h?) + RPT
(h?), and h?=argminh∈H
{E
h′∼ρ
(RDT
(h, h′)− RDS(h, h′)
)}.
� Domain disagreement: disρ(DS ,DT ) = Eh1,h2∼ρ2
[Pr
x∼DS
(h(x) 6=h′(x))−Prx∼DT
(h(x) 6=h′(x))].
Given empirical observations S ∼ (PS)m and T ∼ (DS)m′,
⇒ We want to minimize : BP〈S,T〉(Gρ)def= RPS
(Gρ) + disρ(DS ,DT ) ,where P〈S,T〉 denotes the joint distribution over PS × DT .
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 6 / 10
PAC-Bayesian Learning of Linear Classifier[Germain, Lacasse, Laviolette and Marchand, 2009]
• Let H be a set of linear classifiers hv(x)def= sgn (v · x)
• Consider a prior π0 and a posterior ρw defined as isotropicGaussians respectively centered on vectors 0 and w.
w
Theorem [Langford and Shawe-Taylor, 2002]
For any domain PS ⊆ Rd × Y and any δ ∈ (0, 1], we have,
PrS∼(PS )m
(∀w ∈ Rd : kl
(RS(Gρw )
∥∥RPS (Gρw ))≤ 1
m
[KL(ρw ‖π0) + ln
ξ(m)
δ
])≥ 1−δ.
Trade-off between:� RS(Gρw ) = E
(x,y)∼PS
Φ(y w·x‖x‖
)is the sigmoidal loss
� KL(ρw ‖π0) = 12‖w‖2 is a regularizer
-3 -2 -1 1 2 3a
0.2
0.4
0.6
0.8
1
Φ(a)
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 7 / 10
PAC-Bayesian Domain Adaptation Learning of Linear Classifier
Given empirical observations S ∼ (PS)m and T ∼ (DS)m′,
⇒ We want to minimize : BP〈S,T〉(Gρ)def= RPS
(Gρ) + disρ(DS ,DT ) ,where P〈S,T〉 denotes the joint distribution over PS × DT .
Theorem
For any domain P〈S,T〉 ⊆ Rd × Y × Rd and any δ ∈ (0, 1], we have,
Pr〈S,T〉∼(P〈S,T〉)m
(∀w ∈ Rd : kl
(B∗〈S,T〉
∥∥B∗P〈S,T〉) ≤ 1
m
[2KL(ρw ‖π0) + ln
ξ(m)
δ
])≥ 1−δ ,
where BP〈S,T〉(Gρw ) = RPS (Gρw ) + disρw (DS ,DT ) .
Trade-off between:
� RPS (Gρw ) = E(xs ,y s )∼PS
Φ(y s w·xs‖xs‖
)� disρw (DS ,DT ) = E
(xs ,y s )∼PS
Φdis
(w·xs‖xs‖
)− E
xt∼DT
Φdis
(w·xt‖xt‖
)� KL(ρw ‖π0) = 1
2‖w‖2
-3 -2 -1 1 2 3a
0.2
0.4
0.6
0.8
1
Φ(a)Φdis(a)
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 8 / 10
Preliminary Experimental ResultsBound minimization by gradient descent
Illustration of the decision boundary on 4 rotations angles:
20◦ 30◦ 40◦ 50◦
Rotation angle 20◦ 30◦ 40◦ 50◦
PBGD 99.5 89.8 78.6 60SVM 89.6 76 68.8 60
TSVM 100 78.9 74.6 70.9DASVM 100 78.4 71.6 66.6
DASF 98 92 83 70
DA-PBGD 97.7 97.6 97.4 53.2 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.8
source errortarget error
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 9 / 10
Thank you!
See you at our poster.
Germain et al. (GRAAL, LIF & LaHC) PAC-Bayesian Learning and Domain Adaptation December 7, 2012 10 / 10
